Okay, well, actually Sherlock can probably stay where he is, because (1) the detective, C. Auguste Dupin, lives in Paris, (2) there are only three stories about him, (3) Poe actually influenced Arthur Conan Doyle, and (4) Robert Downey, Jr. is already booked.
Poe may not have outright invented detective fiction, but he certainly left his mark. In "The Purloined Letter" and the two earlier stories, he created a character whose evidence-based detection set the standard for later Sherlockian-style deduction. Poe called it "ratiocination," but we can just call it "thinking like a smart person."
"The Purloined Letter" is the last of the three Dupin stories. It was first published in 1844 in a multi-author collection of stories, poems, and illustrations called The Gift: A Christmas, New Year, and Birthday Present. These kinds of literary annuals topped every gift-giving guide in the nineteenth century, since they were expensive and tasteful—perfect for giving to your girlfriend or putting out to display what great taste you had.
So it makes sense that, while "The Purloined Letter" definitely looks, sounds, and feels like Poe, it's different than his Gothic and Horror stories like "The Tell-Tale Heart" and "The Fall of the House of Usher." Nobody dies, gets buried alive, or tortured. Nobody even goes mad. Instead, we've got a game of wits: no scaredy cats, just smarty-pants.
Speaking of smarty pants, "The Purloined Letter" sparked a famous literary feud between two major players in the literary world: Jacques "Have You Seen My Phallus" Lacan and Jacques "Question Everything" Derrida. (Note that they're both French: While Poe wasn't very popular in America when he was alive, and still has the rep of a third-rate hack, the French were nuts for him.)
In 1957, the psychoanalyst, linguist, literary critic, and very brainy dude Lacan gave a talk on "The Purloined Letter." Almost twenty years later, hugely important literary critic Jacques Derrida fired back an eighty-some-odd-page takedown. The two writers both see the letter as a symbol of different processes of vision, knowing, and desire, so check out "Symbols, Imagery, and Allegory" for more about that.
The point of bringing up these guys is to say that this little story, published in a totally pop-culture format, might have a lot more going on than is immediately apparent. If you get to the end and find yourself wondering what the point of that whole thing was—awesome. You're asking exactly the right question.
A little piece of "The Purloined Letter" is in every detective story you've ever seen, big-screen or small: House, Monk, Psych, Sherlock Holmes, Sherlock Holmes: A Game of Shadows, and, oh yeah, Sherlock … well, we could go on, but we suspect you've heard this story. Brainy, depressed detective uses the sheer force of his intellect to put together clues that everyone else overlooks and solve the mystery.
A combination of the stunningly obvious and the mind-blowingly subtle, the facts of the mystery—what disease the patient is dying of; who killed the vic; what's going on with the royal lady—are always less important than the investigation. After all, no one watches House reruns for medical advice. (That's what the Internet is for.) You watch because everyone loves the thrill of discovery, especially when we don't have to do all the hard thinking for ourselves. It's a little about the madness, but a lot more about the method.
That's why what seems like it's going to be a fun romp through the scandalous halls of the French court turns out to be a much headier exploration of power, corruption, logic, reason, math, poetry, reality, and, you know, nothing at all.
Because "The Purloined Letter" of the title? It might as well be blank. Edgar Allan Poe is so uninterested in the letter that we don't even get a hint at the content. Instead, we have to ask ourselves: if the letter (or disease/mystery/national crisis) doesn't matter, then what does?